Homomorphisms of Fourier algebras and transference results

Abstract

We prove that if $\rho :A(H)\to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $\alpha :G\to H$, then $\rho$ extends to a w*-w* continuous map between the corresponding $L^{\infty }$ algebras if and only if $\alpha$ is an open map. Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $\alpha$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure of $H$, then $(\alpha \times \alpha {)}^{-1}$ preserves sets of compact operator synthesis, and conversely when $\alpha$ is onto. We also prove similar preservation properties for operator Ditkin sets and operator M-sets, obtaining preservation properties for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.

This preprint will be restructured and will be replaced by `Synthetic properties of locally compact groups: preservation and transference' (submitted to arXiv) and `Homomorphisms of Fourier algebras' (in preparation)

References in corpus (1)

• Completely bounded homomorphisms of the Fourier algebra revisited